Solve for $x$ : $ 5|x + 8| + 2 = 3|x + 8| + 10 $
Answer: Subtract $ {3|x + 8|} $ from both sides: $ \begin{eqnarray} 5|x + 8| + 2 &=& 3|x + 8| + 10 \\ \\ { - 3|x + 8|} && { - 3|x + 8|} \\ \\ 2|x + 8| + 2 &=& 10 \end{eqnarray} $ Subtract ${2}$ from both sides: $ \begin{eqnarray} 2|x + 8| + 2 &=& 10 \\ \\ { - 2} &=& { - 2} \\ \\ 2|x + 8| &=& 8 \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{2|x + 8|} {{2}} = \dfrac{8} {{2}} $ Simplify: $ |x + 8| = 4$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 8 = -4 $ or $ x + 8 = 4 $ Solve for the solution where $x + 8$ is negative: $ x + 8 = -4 $ Subtract ${8}$ from both sides: $ \begin{eqnarray} x + 8 &=& -4 \\ \\ {- 8} && {- 8} \\ \\ x &=& -4 - 8 \end{eqnarray} $ $ x = -12 $ Then calculate the solution where $x + 8$ is positive: $ x + 8 = 4 $ Subtract ${8}$ from both sides: $ \begin{eqnarray} x + 8 &=& 4 \\ \\ {- 8} && {- 8} \\ \\ x &=& 4 - 8 \end{eqnarray} $ $ x = -4 $ Thus, the correct answer is $x = -12 $ or $x = -4 $.